Journal of the
Korean Mathematical Society
JKMS

ISSN(Print) 0304-9914 ISSN(Online) 2234-3008

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J. Korean Math. Soc. 2004; 41(3): 479-487

Printed May 1, 2004

Copyright © The Korean Mathematical Society.

On decomposability of finite groups

Ali Reza Ashrafi

University of Kashan

Abstract

Let $G$ be a finite group and $N$ be a normal subgroup of $G$. We denote by $ncc(N)$ the number of conjugacy classes of $N$ in $G$ and $N$ is called $n$-decomposable, if $ncc(N)=n$. Set $K_G = \{ ncc(N) \ | \ N \lhd G \}$. Let $X$ be a non-empty subset of positive integers. A group $G$ is called $X$-decomposable, if $K_G = X$. In this paper we characterise the $\{ 1, 3, 4 \}$-decomposable finite non-perfect groups. We prove that such a group is isomorphic to $SmallGroup$ $(36,9)$, the $9^{th}$ group of order 36 in the small group library of GAP, a metabelian group of order $2^n(2^{\frac{n-1}{2}}-1)$, in which $n$ is odd positive integer and $2^{\frac{n-1}{2}}-1$ is a Mersenne prime or a metabelian group of order $2^n(2^{\frac{n}{3}}-1)$, where $3 | n$ and $2^{\frac{n}{3}}-1$ is a Mersenne prime. Moreover, we calculate the set $K_G$, for some finite group $G$.

Keywords: finite group, $n$-decomposable subgroup, conjugacy class

MSC numbers: Primary 20E34, 20D10